The number of maximal independent sets of the n-cycle graph
Cn is
known to be the nth term of the Perrin sequence. The action of the
automorphism group of Cn
on the family of these maximal independent
sets partitions this family into disjoint orbits, which represent the
non-isomorphic (i.e., defined up to a rotation and a reflection)
maximal independent sets. We provide exact formulas for the total
number of orbits and the number of orbits having a given number of
isomorphic representatives. We also provide exact formulas for the
total number of unlabeled (i.e., defined up to a rotation) maximal
independent sets and the number of unlabeled maximal independent sets
having a given number of isomorphic representatives. It turns out that
these formulas involve both the Perrin and Padovan sequences.